Search

Objective To correct for image distortions produced by standard Fourier reconstruction techniques on low field permanent magnet MRI systems with strong $${B}_{0}$$ B 0 inhomogeneity and gradient field nonlinearities. Materials and methods Conventional image distortion correction algorithms require accurate $${\Delta B}_{0}$$ Δ B 0 maps which are not possible to acquire directly when the $${B}_{0}$$ B 0 inhomogeneities also produce significant image distortions. Here we use a readout gradient time-shift in a TSE sequence to encode the $${B}_{0}$$ B 0 field inhomogeneities in the k-space signals. Using a non-shifted and a shifted acquisition as input, $$\Delta {B}_{0}$$ Δ B 0 maps and images were reconstructed in an iterative manner. In each iteration, $$\Delta {B}_{0}$$ Δ B 0 maps were reconstructed from the phase difference using Tikhonov regularization, while images were reconstructed using either conjugate phase reconstruction (CPR) or model-based (MB) image reconstruction, taking the reconstructed field map into account. MB reconstructions were, furthermore, combined with compressed sensing (CS) to show the flexibility of this approach towards undersampling. These methods were compared to the standard fast Fourier transform (FFT) image reconstruction approach in simulations and measurements. Distortions due to gradient nonlinearities were corrected in CPR and MB using simulated gradient maps. Results Simulation results show that for moderate field inhomogeneities and gradient nonlinearities, $$\Delta {B}_{0}$$ Δ B 0 maps and images reconstructed using iterative CPR result in comparable quality to that for iterative MB reconstructions. However, for stronger inhomogeneities, iterative MB reconstruction outperforms iterative CPR in terms of signal intensity correction. Combining MB with CS, similar image and $$\Delta {B}_{0}$$ Δ B 0 map quality can be obtained without a scan time penalty. These findings were confirmed by experimental results. Discussion In case of $${B}_{0}$$ B 0 inhomogeneities in the order of kHz, iterative MB reconstructions can help to improve both image quality and $$\Delta {B}_{0}$$ Δ B 0 map estimation.

The paper presents a novel method for interpolating rotations based on the non-Abelian Kuramoto model on sphere S3. The algorithm, introduced in this paper, finds the shortest and most direct path between two rotations. We have discovered that it gives approximately the same results as a Spherical Linear Interpolation algorithm. Simulation results of our algorithm are visualized on S2 using Hopf fibration. In addition, in order to gain a better insight, we have provided one short video illustrating the rotation of an object between two positions.

Model estimation is an important issue in signal processing since it enables to characterize the signal and to obtain some parameters providing information about its shape. An example of a model widely used in various fields of science is the sum of exponentially damped sinusoids. In this work, a new method for parameters estimation of such a model is proposed. This method uses nonlinear least squares together with Fourier transform to estimate parameters of the sum of damped sinusoids. For model order selection, a bootstrap method is utilized. First, the presented method is verified on simulated signals. The efficacy of model order selection is assessed for signals with a different set value of signal-to-noise (SNR) ratio. It is shown that for the models with orders from 2 to 7, the order was estimated correctly in all trials for SNR greater than 20 dB. If a proper model order is selected, the set values of signal parameters are included in the confidence intervals for parameters values, determined by the nonlinear least squares method. Afterward, the described method is used for modeling the event-related potential in magnetoencephalography (MEG) signal. The method gives good fitting results with \(\text {R}^{2}\) values greater than 0.9, showing its potential in MEG signals modeling and the assessment of brain activity.

In this paper, mathematical modelling based on strip theory of the Hydrodynamically Actuated Manipulator (HAM) is proposed to use it for analysis and control synthesis. A set of Nonlinear Ordinary Differential Equations (ODEs) describes the behaviour of the HAM. First, considering the Euler-Lagrange methodology, the motion of the \(R \, \bot \, R \, \parallel \, P\) manipulator is obtained. Then, a Newtonian approach is under study to add the effects of a fluid surrounding the 3-DOF manipulator. In order to describe this interaction, each link of the manipulator affected by hydrodynamic forces is discretized on small elements. Their contribution is added to obtain the net acceleration of each link of the HAM. In order to qualitatively validate the model, a numerical simulation are under study. The experiment consists of a comparison between the Lagrangian of the 3-DOF manipulator and its motion when is affected by each added hydrodynamic force.

In this chapter, new results on the observer-based control problem for fractional-order systems are presented. The two control philosophies, which are the “stabilization” and the “model-reference control”, are investigated. First, dealing with the stabilization problem, an adaptive stabilization scheme for a class of uncertain nonlinear fractional systems is detailed. Second, the model-reference control problem is investigated. A novel and advantageous model-reference control design for linear fractional-order systems is presented.

It is critical to reduce the electric energy consumption of information systems, especially clusters of servers. In order to reduce the energy consumption, we have to select a virtual machine on an energy-efficient server to perform an application process so that total energy consumption of servers can be reduced. Here, we have to estimate how much energy a server consumes to perform processes. In the simple estimation (SP) algorithm previously proposed, every current process is assumed to have the same computation residue. The execution time of each process and energy consumption of the server are underestimated while it takes short time to do the estimation. In this paper, we newly propose a novel monotonically increasing (MI) algorithm to estimate energy consumption of a host server to perform application processes. Here, current processes \(p_1\), \(\ldots \), \(p_{n_t}\) on a server \(s_t\) are assumed to be totally ordered so that the computation residue \({RP}_i\) of a process \(p_i\) is \({RP}_1 \cdot {\alpha }^{i-1}\) larger than a process \(p_{i-1}\) (i \(=\) \(1, \ldots , n_t\)). In the evaluation, we show the execution time and energy consumption of a server obtained by the MI algorithm are only one [%] different from the simulation results.

In this paper, we concentrate on the two-dimensional \(\alpha \)-stable autoregressive model of order 1. For this model, we analyze the auto-dependence functions applied to describe the interdependence of the time series components considered separately as the one-dimensional processes. Since the classical second moment-based dependence measures are not defined in the \(\alpha -\)stable case, we examine here the auto-codifference and the auto-covariation functions. The main result of this paper is the derivation of the formulas which describe the asymptotics of the considered measures of interdependence. The theoretical results are supported by a simulation study.

In the frame of its electro-nuclear activity, EDF has to deal with nuclear wastes. Containment of waste packages must be demonstrated. The latter is strongly influenced by the delayed strains of cement materials that can lead to cracking. Moreover, the permeability of concrete itself is vastly dependent on its saturation degree. For these reasons, a good prediction of strains and leak-tightness of concrete containment must start with a good estimate of the moisture profile on concrete. In this framework a large panel of experiments and studies are performed in order to demonstrate the package containment. The temperature experienced by waste packages is closed to 50 \({}^{\circ }\text {C}\) in the concrete shell and 60 \({}^{\circ }\text {C}\) in the filling slurry. Moreover, traditionally, the identification of the parameters of concrete drying constitutive laws is based on mass measurement of samples exposed to a constant humidity and temperature. However, it has been observed for different drying laws that this methods leads to a multiplicity of solutions, potentially corresponding to different moisture profiles and also to different extrapolations of the mass-loss beyond the end-time of the experiment used for the identification [1]. As an attempt to calibrate drying model, this study is interested in both obtaining drying experimental data in temperature and also original experimental data set that can discriminate between numerical parameter sets. Thus, data regarding relative mass variation, desorption isotherm at various temperature and relative humidity are carried out. Finally an easy way of achieving cup test is presented in order to assess water vapor permeability for different RH levels.

An interconnection network is a programmable system that serves to transport data or messages amongst network components and/or terminals. A network’s topology is typically modeled by a graph. A path of order k in a graph G is a sequence of k distinct vertices, denoted by \(P_k = \langle v_1,v_2,\cdots ,v_k\rangle \), in which any two consecutive vertices are adjacent. The connectivity is a classic index to assess the level of network reliability and fault tolerance. For \(k \ge 2\), a set F of vertex subsets of G is a \(P_k\)-cut if \(G-F\) is disconnected, and each element of F happens to induce a \(P_k\)-subgraph in G. A connected graph G is super \(P_k\)-connected if the smallest component of \(G-F\) is a singleton for every minimum \(P_k\)-cut F of G. A network with smaller diameter can reduce its communication delay in a worst-case perspective. The crossed cube \(CQ_n\) is a hypercube variant whose diameter is about one half of that of the hypercube. This paper is inspired to discover whether \(CQ_n\) is super \(P_k\)-connected for \(k=2,3,4\).

Given a locally finite set $$X \subseteq {{\mathbb {R}}}^d$$ X ⊆ R d and an integer $$k \ge 0$$ k ≥ 0 , we consider the function $${\mathbf{w}_{k}^{}} :{\mathrm{Del}_{k}{({X})}} \rightarrow {{\mathbb {R}}}$$ w k : Del k ( X ) → R on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.

Consider basic operations with qubits. A quantum gate effect on qubit \(\left| {\psi }\right\rangle \) occurs by applying the quantum-mechanical operator, e.g. \(U\left| {\psi }\right\rangle \) [1, 2, 3]. Operators can be represented as unitary matrices. In particular, the evolution of a single qubit is described by a unitary matrix of size \(2\times 2\). The consistent application of a number of operators \(U_1\), \(U_2\), ..., \(U_n\) to one qubit is equivalent to the effect of some operator W in the form $$W\left| {\psi }\right\rangle = U_n(U_{n-1}(\dots (U_2(U_1\left| {\psi }\right\rangle ))\ldots )) = (U_1U_2\ldots {}U_n)\left| {\psi }\right\rangle .$$ Its matrix \(M_W\) is a product of matrices of \(U_i\), \(i=1,2,\dots ,n\), in the reverse order [4]: $$M_W = M_{U_n}M_{U_{n-1}}\dots M_{U_1}.$$ Such an operator W is called a product of operators \(U_1\), \(U_2\), ..., \(U_n\). Due to the noncommutativity of the matrix multiplication operation, the order in which quantum gates are applied is generally important.

Aperiodic order is a relatively young area of mathematics with connections to many other fields, including discrete geometry, harmonic analysis, dynamical systems, algebra, combinatorics and, above all, number theory. In fact, numbertheoretic methods and results are present in practically all of these connections. It was one aim of this workshop to review, strengthen and foster these connections.

Set-theoretic solutions of the Yang–Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution (X, r) consists of a set X and a bijective map \(r:X\times X\rightarrow X\times X\) which satisfies the braid relations. In this work we study the braided group \(G=G(X,r)\) of an involutive square-free solution (X, r) of finite order n and cyclic index \(p=p(X,r)\) and the group algebra \(\mathbf{k} [G]\) over a field \(\mathbf{k} \). We show that G contains an invariant subgroup \({\mathcal {F}}_p\) of finite index \(p^n\) which is isomorphic to the free abelian group of rank n. We describe explicitly the quotient braided group \(\widetilde{G}=G/{\mathcal {F}}_p\) of order \(p^n\) and show that X is embedded in \(\widetilde{G}\). We prove that the group algebra \(\mathbf{k} [G]\) is a free left (resp. right) module of finite rank \(p^n\) over its commutative subalgebra \(\mathbf{k} [{\mathcal {F}}_p]\) and give an explicit free basis. The center of \(\mathbf{k} [G]\) contains the subalgebra of symmetric polynomials in \(\mathbf{k} [x_1^p, \cdots , x_n^p]\). Classical results on group rings imply that \(\mathbf{k} [G]\) is a left (and right) Noetherian domain of finite global dimension.

In this chapter we consider the arithmetics of arbitrary large integers. Integer arithmetic is the backbone of every computer algebra system, hence we would like to discuss this topic in some detail. In order to be able to compute with integers, they must be suitably represented. In the history of Mathematics, several representations of integers have been introduced, two of which are still in use: the roman and the arabic number systems.

We express the Kirchhoff wave equation in terms of classic field theory. This permits us to introduce the spontaneous symmetry breaking phenomenon in the study of linear structures, such as strings in order to investigate the existence of solitons solutions. We find ϕ4 solitons in the space of spatial gradient of lateral displacement of a string. This helps us detect stable states in deformations of strings.

The goal of this chapter is to derive an a posteriori error estimate for second-order elliptic PDEs approximated by \(H^1\)-conforming finite elements. Such an estimate is an upper bound on the approximation error that can be computed by using only the discrete solution and the problem data. It can serve the twofold purpose of judging the quality of the discrete solution and of guiding an adaptive procedure that modifies the discretization iteratively in order to diminish the approximation error. For the purpose of mesh adaptation, the error estimate should be a sum of local contributions (usually called indicators) that can be used to mark those cells requiring further refinement at the next iteration of the adaptive procedure. It is also important that the indicators represent a local lower bound on the error.

Here are introduced left and right Riemann–Liouville generalized fractional radial integral operators of variable order over a spherical shell. Also are introduced left and right weighted Caputo type generalized fractional radial derivatives of variable order over a spherical shell.

The article deals with a Boolean valued approach to some algebraic problems arising from functional analysis. The main results are as follows. (1) A universally complete vector lattice without locally one-dimensional bands contains an infinite direct sum of order dense sublattices each of which is a band preserving linear isomorphic (but not lattice isomorphic) copy of the whole lattice. (2) Every separated injective module over a semiprime rationally complete commutative ring admits a direct sum decomposition with homogeneous summands. (3) A semiprime rationally complete commutative ring properly embedded in a ring with projections K is a homogeneity ring of an additive mapping between appropriate K-modules.

In this paper, to reduce impulsive noise in color images we propose an extension of the Median Redescending M-Estimator. For that purpose, a multitasking approach was developed such as a multi-core processing in order to reduce in parallel the noise on R, G and B color channels. With this paradigm, an acceleration up to three times can be guaranteed compared to the sequential paradigm, while having the ability to reduce corrupted data up to densities of \(80\%\) of fixed-value and \(40\%\) of random-value impulsive noises, guaranteeing the preservation of edges. The effectiveness of our proposal is verified by quantitative and qualitative results.

We investigate the standard context, denoted by \(\mathbb {K}\left( \mathcal {L}_{n}\right) \), of the lattice \(\mathcal {L}_{n}\) of partitions of a positive integer n under the dominance order. Motivated by the discrete dynamical model to study integer partitions by Latapy and Duong Phan and by the characterization of the supremum and (infimum) irreducible partitions of n by Brylawski, we show how to construct the join-irreducible elements of \(\mathcal {L}_{n+1}\) from \(\mathcal {L}_{n}\). We employ this construction to count the number of join-irreducible elements of \(\mathcal {L}_{n}\), and confirm that the number of objects (and attributes) of \(\mathbb {K}\left( \mathcal {L}_{n}\right) \) has order \(\varTheta (n^2)\). We also discuss the embeddability of \(\mathbb {K}\left( \mathcal {L}_{n}\right) \) into \(\mathbb {K}\left( \mathcal {L}_{n+1}\right) \) with special emphasis on \(n=9\).

The aim of this manuscript is to investigate analytic and numerical solutions of space–time fractional diffusion wave equations with different fractional order (\(\alpha \) and \(\beta \)). After deriving analytic solution, an implicit unconditional stable finite difference method for solving space-time fractional diffusion wave equations is proposed. The Gerschgorin theorem is used to study the stability and convergence of the method. Furthermore, the behavior of the error is examined to verify the order of convergence by numerical example.

In Part II of this text, we investigated arithmetic and algebraic properties of quaternion orders and ideals. Now in Part III, continuing this investigation we turn to the use of analytic methods. In this first introductory chapter, we restrict to a special case and consider zeta functions of quaternion orders over the rationals; as an application, we obtain a formula for the mass of the class set of a definite quaternion order.

In the space \(\mathrm {H}_n(q^2)\) of Hermitian matrices over \(\mathbb {F}_{q^2}\) of order n we can define a d-code as subset \(\mathrm {C}\) of \(\mathrm {H}_n(q^2)\) such that \(\mathrm {rk}(A-B)\ge d\) for every \(A, B \in \mathrm {C}\) with \(A\ne B\). In \(\mathrm {H}_n(q^2)\) we give two possible equivalence definitions: 1) the ones coming from the maps that preserve the rank in \(\mathbb {F}_{q^2}^{n\times n}\); 2) the ones that come from restricting to those maps preserving both the rank and the space \(\mathrm {H}_n(q^2)\).

Let \(\mathrm{PG}(n,q)\) be the n-dimensional projective space over the finite field \({\mathbb F}_q\). A spread in \(\mathrm{PG}(n,q)\) is a set of lines which partition the point set. A partition of the lines of the projective space by spreads is called a parallelism. The study of parallelisms is motivated by their numerous relations and applications. We construct 8958 new nonisomorphic parallelisms of \(\mathrm{PG}(3,5)\). They are invariant under cyclic automorphism groups of order 8. Some of their interesting properties are discussed. We use the system for computational discrete algebra GAP as well as our own MPI-based software written in C++.

We investigate the problem of drawing two posets of the same ground set so that one is drawn from left to right and the other one is drawn from the bottom up. The input to this problem is a directed graph \(G = (V, E)\) and two sets X, Y with \(X \cup Y = E\), each of which can be interpreted as a partial order of V. The task is to find a planar drawing of G such that each directed edge in X is drawn as an x-monotone edge, and each directed edge in Y is drawn as a y-monotone edge. Such a drawing is called an xy-planar drawing.

The loose core in hypergraphs is a structure inspired by loose cycles which mirrors the close relationship between 2-cores and cycles in graphs. We prove that the order of the loose core undergoes a phase transition at a certain critical threshold in the r-uniform binomial random hypergraph \(H^r(n,p)\) for every \(r\ge 3\). We also determine the asymptotic number of vertices and edges in the loose core of \(H^r(n,p)\). Furthermore we obtain an improved upper bound on the length of the longest loose cycle in \(H^r(n,p)\).

The Polyline Bundle Simplification (PBS) problem is a generalization of the classical polyline simplification problem. Given a set of polylines, which may share line segments and points, PBS asks for the smallest consistent simplification of these polylines with respect to a given distance threshold. Here, consistent means that each point is either kept in or discarded from all polylines containing it. In previous work, it was proven that PBS is NP-hard to approximate within a factor of \(n^{\frac{1}{3}-\varepsilon }\) for any \(\varepsilon > 0\) where n denotes the number of points in the input. This hardness result holds even for two polylines. In this paper we first study the practically relevant setting of planar inputs. While for many combinatorial optimization problems the restriction to planar settings makes the problem substantially easier, we show that the inapproximability bound known for general inputs continues to hold even for planar inputs. We proceed with the interesting special case of PBS where the polylines form a rooted tree. Such tree bundles naturally arise in the context of movement data visualization. We prove that optimal simplifications of these tree bundles can be computed in \(\mathcal {O}(n^3)\) for the Frechet distance and in \(\mathcal {O}(n^2)\) for the Hausdorff distance (which both match the computation time for single polylines). Furthermore, we present a greedy heuristic that allows to decompose polyline bundles into tree bundles in order to make our exact algorithm for trees useful on general inputs. The applicability of our approaches is demonstrated in an experimental evaluation on real-world data.

In this chapter, we give a survey of the most basic results on Lyapunov-type inequalities for second-order linear differential equations of the form $$\displaystyle x^{\prime \prime }+q(t)x=0 $$ and sketch some important developments related to this type of inequalities.

Packet error in IEEE 802.15.6 MAC protocol is one of the main sources that degrades the performance and its inconsistency. This protocol does not have the ability to control the packet errors which is caused by the transmission failures or collisions. To alleviate the issues presented in the traditional protocol, this paper aims to develop an analytical model \((p_{r})\) to study the effects of packet errors and validate the performance of the IEEE 802.15.6 MAC protocol using Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) mechanism. The \(p_{r}\)-model decreases the packet error rate by incorporating the optimum number of Retry limit (\(R_l\)) and Contention Window (CW) size for all User Priority (UP) nodes. Therefore, this model can be deliberated as the first-ever analytical model for optimizing the UP and \(R_{l}\) assignment in order to improve the overall performance of the Wireless Body Area Network system. The simulation results suggest that minimizing transmission failure probability has more effect on saturation throughput than multiplying the CW size after even number of failures.

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees the vector of coefficients of the polynomial as a word on a ternary alphabet $\{-1,0 ,+1\}$. It designs an efficient algorithm that computes a compact representation of this word. This algorithm is of linear time with respect to the size of the output, and, thus, optimal. This approach allows to recover known properties of coefficients of binary cyclotomic polynomials, and extends to the case of polynomials associated with numerical semi-groups of dimension 2.

String metrics play an important role in various computational tasks. In this paper, we focus on a property shared by a subset of string rearrangement metrics, called pseudo-locality. This subset includes, for example, the swap, interchange and parallel-interchange distances. Intuitively, operators of pseudo-local string metrics have a bounded effect on the Hamming distance between the string prior to the operation and after it. The goal of this paper is to examine how to exploit pseudo-locality in order to transform tools derived for the Hamming distance in order to derive tools for other pseudo-local metrics. Specifically, we demonstrate such a way to exploit pseudo-locality of the interchange distance combined with additional techniques to derive: 1. The first efficient approximate nearest-neighbour (ANN) search data structure for the interchange distance, which is \(\mathcal {NP}\)-hard to compute for general strings even for binary alphabets. 2. The first linear time algorithm to compute approximate pattern matching with interchanges, which is a vast improvement from the \(\varTheta (nm)\) known algorithm.

In this paper an new approach combining 2-bit and 3-bit BCH error correction with fast and simple error detection for errors of higher order is presented. Under the assumption that a 2-bit error or 3-bit error occurred, the corresponding correction bits are determined using the syndrome components \(s_1, s_3\) for 2-bit errors and \(s_1, s_3, s_5\) for 3-bit errors. These correction bits are used to calculate higher syndrome components up to \(s_{2T-1}\), which are compared to the actual corresponding syndrome components. If the syndrome components match, a 2-bit error or 3-bit error was detected. In the case of a syndrome mismatch, an error other than a 2-bit error or 3-bit error occurred and was detected. The proposed method provides a simple way to differentiate between 2-bit errors (3-bit errors) and errors of higher order.

In this chapter, we prove the Eichler mass number for a definite quaternion order over a totally real field using classical analytic methods.

The standard mass-action kinetics modeling of the dynamics of biochemical reaction networks gives rise to systems of ordinary polynomial differential equations with (in general unknown) parameters. Attempts to explore the parameter space in order to predict properties of the associated systems challenge the standard current computational tools because even for moderately small networks we need to study families of polynomials with many variables and many parameters. These polynomials have a combinatorial structure that comes from the digraph of reactions. We show that different techniques can be strengthened and applied for biochemical networks with special structure.

In this paper, we analyze gradient-free methods with one-point feedback for stochastic saddle point problems \(\min _{x}\max _{y} \varphi (x, y)\). For non-smooth and smooth cases, we present an analysis in a general geometric setup with the arbitrary Bregman divergence. For problems with higher order smoothness, the analysis is carried out only in the Euclidean case. The estimates we have obtained repeat the best currently known estimates of gradient-free methods with one-point feedback for problems of imagining a convex or strongly convex function. The paper uses three main approaches to recovering the gradient through finite differences: standard with a random direction, as well as its modifications with kernels and residual feedback. We also provide experiments to compare these approaches for the matrix game.

Random sets models derived from truncated RF, with an application to Gaussian random functions, are introduced. Covariance and order m central correlation functions are given. The model is illustrated by examples of application.

In this paper, we study the computational complexity of the commutative determinant polynomial computed by a class of set-multilinear circuits which we call regular set-multilinear circuits. Regular set-multilinear circuits are commutative circuits with a restriction on the order in which they can compute polynomials. A regular circuit can be seen as the commutative analogue of the ordered circuit defined by Hrubes, Wigderson and Yehudayoff [5]. We show that if the commutative determinant polynomial has small representation in the sum of constantly many regular set-multilinear circuits, then the commutative permanent polynomial also has a small arithmetic circuit.

3D depth computation from stereo data has been one of the most researched topics in computer vision. While state-of-art approaches have flourished over time, reconstruction of transparent materials is still considered an open problem. Based on 3D light field data we propose a method to obtain smooth and consistent double-layer estimates of scenes with transparent materials. Our novel approach robustly combines estimates from models with different layer hypotheses in a cost volume with subsequent minimization of a joint second order \(\mathrm {TGV}\) energy on two depth layers. Additionally we showcase the results of our approach on objects from common inspection use-cases in an industrial setting and compare our work to related methods.

In this paper we investigate the cognitive potential of Derivative Meaning—defined in terms of Abstraction Tracking (Shimojima 2015)—in order to characterise various families of Aristotelian diagrams. In a first part we consider the notion of subdiagrams—i.e. uniform triangles of contrariety relations and implication relations—inside Aristotelian hexagons and octagons. In a second part we look at different strategies for embedding complete Aristotelian diagrams—i.e. classical and degenerate squares—into Aristotelian hexagons and octagons.

During the 1960s considerable work was done in order to understand the meaning of “epimorphism”. The notion plays an important role in categories of rings where the abstract category-theoretic meaning is now of common use.

\(\varphi \)-families of probability distributions are a general representation to deformed exponential models which brings interesting insight on the geometrical aspects of the distribution family. In the \(\varphi \)-family, the analogue of the cumulant-generating function is a normalizing function. This function plays an important role in the statistical model and we investigate the behaviour of the function near the boundary of the domain of the parametrization in order to provide the precise conditions of validity of the \(\varphi \)- family model. We discuss the conditions for existence of the statistical model when considering \(\varphi \)-functions and the required conditions for the normalization function so we can provide a complete understanding about the investigated family of probability distribution.

In this paper, we construct and analyse C1 quartic interpolating splines on type-1 triangulations, approximating regularly distributed data. This is achieved by defining the associated Bernstein-Bezier coefficients from point values of the function to be approximated in such a way that C1 regularity is obtained for enough regular functions as well as the optimal order of approximation. We construct such interpolating splines by combining a quasi-interpolating spline with one step of an interpolatory subdivision scheme. Numerical tests confirming the theoretical results are provided.

It is shown that the approximation rate of continuous functions by Vallee Poussin means \(V_{n,m}^\alpha (f)\) of special series partial sums (α is a parameter in special series construction) is of the order of the best approximation provided \(\frac {1}{2}

While you are reading this thesis, at every instant of time countless particles of the surrounding air hit your skin due to their irregular thermal motion. There are of the order of \({\sim } 10^{23}\) molecules in every liter of air [1, 2].

Almost 60 years ago Zolotarev proved a duality result which relates an α-stable density for α ∈ (1, 2) to the density of a \(\frac 1{\alpha }\)-stable distribution on the positive real line. In recent years Zolotarev duality was the key to show space-time duality for fractional diffusions stating that certain heat-type fractional equations with a negative fractional derivative of order α in space are equivalent to corresponding time-fractional differential equations of order \(\frac 1{\alpha }\). The point source solutions of the former are given by negatively skewed α-stable densities, whereas the latter are solved by densities of corresponding inverse \(\frac 1{\alpha }\)-stable subordinators. We review this space-time duality and take it as a recipe for a previously unknown generalization from the stable to the semistable situation.

Titanium alloys are substantially used in the aerospace industry due to their high strength to weight ratio. During turning of titanium alloys, obtaining smooth surface with dimensional accuracy is a requirement in machining of parts. The quality of machining is a function of machining parameters used in turning. Twenty-seven experiments were conducted with different combination of cutting speed(Vc), Feed rate (f) and Depth of cut (ap) and the corresponding surface roughness (Ra) was measured. Response surface methodology (RSM) was firstly developed with the experimental data and Genetic algorithm(GA) was then applied to minimize the surface roughness. Validation experiments showed that the error rate was 4.27%. In order to minimize the error rate, a systematic approach was implemented to develop optimal Artificial neural network (ANN) model with the contemplation of effect of ANN architecture and activation functions on the prediction. The predicted ANN data were further used to develop revised RSM model. The later prediction model and the respective optimization resulted the best cutting parameters for achieving the minimum Ra. The predicted surface roughness from GA is \( R_{a} = 1.42 \mu {\text{m}} \) for the optimum turning condition of \( V_{c} = 148\,{\text{m/minute}},\, f = 0.1\,{\text{mm/rev}}, ap = 0.5 \,{\text{mm}} \). The error rate between the predicted results and the validation results is only 2.27%. The proposed model can be used practically in the turning of titanium alloys.

The magnetic moment of spin-1/2 fermion with mass m and charge e is predicted as \(\mu = e/2m\) from the Dirac equation. The fermion precesses in magnetic field (B) with angular velocity \(\omega _p = eB/m\) due to the magnetic moment. On the other hand, the cyclotron frequency of the fermion moving in the same magnetic field is \(\omega _c = eB/m\). Since \(\omega _c = \omega _p\), the helicity (spin component of the momentum direction) of the fermion conserves in any magnetic field. However, it is observed that there is actually a slight difference between \(\omega _c\) and \(\omega _p\). The origin of the difference is that the higher order diagrams add very small spin-flipping transition amplitude to the original one and change \(\omega _p\). This effect is important because we can look new physics out through the additional transition amplitude.

E-functions are entire functions with algebraic Taylor coefficients at the origin satisfying certain arithmetic conditions, and solutions of linear differential equations with coefficients in \(\overline{\mathbb Q}(z)\); they naturally generalize the exponential function. Siegel and Shidlovsky proved a deep transcendence result for their values at algebraic points. Since then, a lot of work has been devoted to apply their theorem to special E-functions, in particular the hypergeometric ones. In fact, Siegel asked whether any E-function can be expressed as a polynomial in z and generalized confluent hypergeometric series. As a first positive step, Shidlovsky proved that E-functions with order of the differential equation equal to 1 are in \(\overline{\mathbb Q}[z]e^{\overline{\mathbb Q}z}\). In this paper, we give a new proof of a result of Gorelov that any E-function (in the strict sense) with order \(\le 2\) can be written in the form predicted by Siegel with confluent hypergeometric functions \({}_1F_1[\alpha ;\beta ;\lambda z]\) for suitable \(\alpha , \beta \in \mathbb Q\) and \(\lambda \in \overline{\mathbb Q}\). Gorelov’s result is in fact more general as it holds for E-functions in the large sense. Our proof makes use of Andre’s results on the singularities of the minimal differential equations satisfied by E-functions, together with a rigidity criterion for (irregular) differential systems recently obtained by Bloch-Esnault and Arinkin. An ad-hoc version of this criterion had already been used by Katz in his study of confluent hypergeometric series. Siegel’s question remains unanswered for orders \(\ge 3\).

For the single index model y = f(βτx, 𝜖) with Gaussian design, where f is unknown and β is a sparse p-dimensional unit vector with at most s nonzero entries, we are interested in testing the null hypothesis that β, when viewed as a whole vector, is zero against the alternative that some entries of β is nonzero, with n i.i.d observations. Assuming that \(var(\mathbb {E}[\boldsymbol {x} \mid y])\) is non-vanishing, we define the generalized signal-to-noise ratio (gSNR) λ of the model as the unique non-zero eigenvalue of \(var(\mathbb {E}[\boldsymbol {x} \mid y])\). We show that if \(s^{2}\log ^2(p)\wedge p\) is of a smaller order of n, denoted as \(s^{2}\log ^2(p)\wedge p\prec n\), one can detect the existence of signals if and only if gSNR\(\succ \frac {p^{1/2}}{n}\wedge \frac {s\log (p)}{n}\). Furthermore, if the noise is additive (i.e., y = f(βτx) + 𝜖), one can detect the existence of the signal if and only if gSNR\(\succ \frac {p^{1/2}}{n}\wedge \frac {s\log (p)}{n} \wedge \frac {1}{\sqrt {n}}\). It is rather surprising that the detection boundary for the single index model with additive noise matches that for linear regression models. These results pave the road for thorough theoretical analysis of single/multiple index models in high dimensions.